Multilinear map

Linear map of two variables.

More formally, given 3 vector spaces X, Y, Z over a single field, a bilinear map is a function from: that is linear on the first two arguments from X and Y, i.e.: Note that the definition only makes sense if all three vector spaces are over the same field, because linearity can mix up each of them.

The most important example by far is the dot product from ${\mathbb{R}}^n\times {\mathbb{R}}^n\to \mathbb{R}$, which is more specifically also a symmetric bilinear form.

Analogous to a linear form, a bilinear form is a Bilinear map where the image is the underlying field of the vector space, e.g. ${\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R}$.

Some definitions require both of the input spaces to be the same, e.g. ${\mathbb{R}}^n\times {\mathbb{R}}^n\to \mathbb{R}$, but it doesn't make much different in general.

The most important example of a bilinear form is the dot product. It is only defined if both the input spaces are the same.

As usual, it is useful to think about how a bilinear form looks like in terms of vectors and matrices.

Unlike a linear form, which was a vector, because it has two inputs, the bilinear form is represented by a matrix $M$ which encodes the value for each possible pair of basis vectors.

In terms of that matrix, the form $B(x,y)$ is then given by:

If $C$ is the change of basis matrix, then the matrix representation of a bilinear form $M$ that looked like: then the matrix in the new basis is: Sylvester's law of inertia then tells us that the number of positive, negative and 0 eigenvalues of both of those matrices is the same.

Proof: the value of a given bilinear form cannot change due to a change of bases, since the bilinear form is just a function, and does not depend on the choice of basis. The only thing that change is the matrix representation of the form. Therefore, we must have: and in the new basis: and so since:

Related:

See form.

Analogous to a linear form, a multilinear form is a Multilinear map where the image is the underlying field of the vector space, e.g. ${\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}\times {\mathbb{R}}^{n_2}\to \mathbb{R}$.

Subcase of symmetric multilinear map:

Requires the two inputs $x$ and $y$ to be in the same vector space of course.

The most important example is the dot product, which is also a positive definite symmetric bilinear form.

symmetric bilinear maps that is also a bilinear form.

Like the matrix representation of a bilinear form, it is a matrix, but now the matrix has to be a symmetric matrix.

We can then immediately see that the matrix is symmetric, then so is the form. We have: But because $B(x,y)$ is a scalar, we have: and:

The complex number analogue of a symmetric bilinear form.

The prototypical example of it is the complex dot product.

Note that this form is neither strictly symmetric, it satisfies: where the over bar indicates the complex conjugate, nor is it linear for complex scalar multiplication on the second argument.

Bibliography:

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A Hermitian matrix.

Multivariate polynomial where each term has degree 2, e.g.: is a quadratic form because each term has degree 2:but e.g.: is not because the term $x^3$ has degree 3.

More generally for any number of variables it can be written as:

There is a 1-to-1 relationship between quadratic forms and symmetric bilinear forms. In matrix representation, this can be written as: where $\vec{x}$ contains each of the variabes of the form, e.g. for 2 variables:

Strictly speaking, the associated bilinear form would not need to be a symmetric bilinear form, at least for the real numbers or complex numbers which are commutative. E.g.: But that same matrix could also be written in symmetric form as: so why not I guess, its simpler/more restricted.

Symmetric bilinear form that is also positive definite, i.e.:

A positive definite matrix that is also a symmetric matrix.

Subcase of antisymmetric multilinear map:

Skew-symmetric bilinear map that is also a bilinear form.

Same value if you swap any input arguments.

Change sign if you swap two input values.

Implies antisymmetric multilinear map.